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THE FREQUENCIES of VARIOUS INTERPRETATIONS of the DEFINITE INTEGRAL in a GENERAL STUDENT POPULATION Steven R

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THE FREQUENCIES of VARIOUS INTERPRETATIONS of the DEFINITE INTEGRAL in a GENERAL STUDENT POPULATION Steven R THE FREQUENCIES OF VARIOUS INTERPRETATIONS OF THE DEFINITE INTEGRAL IN A GENERAL STUDENT POPULATION Steven R. Jones Brigham Young University Student understanding of integration has become a topic of recent interest in calculus research. Studies have shown that certain interpretations of the definite integral, such as the area under a curve or the values of an anti-derivative, are less productive in making sense of contextualized integrals, while on the other hand understanding the integral as a Riemann sum or as “adding up pieces” is highly productive for contextualized integrals. This report investigates the frequency of these three conceptualizations in a general calculus student population. Data from student responses show a high prevalence of area and anti-derivative ideas and a very low occurrence of summation ideas. This distribution held even for students whose calculus instructors focused on Riemann sums while introducing the definite integral. INTRODUCTION First-year calculus has received much attention in mathematics education in recent years due to its significance in science, technology, engineering, and mathematics (STEM) fields. In particular, the calculus concept of the definite integral has become a current topic of interest among mathematics education researchers (e.g. Black & Wittmann, 2007; Hall, 2010; Jones, 2013; Sealey & Oehrtman, 2007; Thompson & Silverman, 2008). The integral is an important topic to investigate because it is commonly used in subsequent mathematics courses (see Brown & Churchill, 2008; Fitzpatrick, 2006) and provides the foundation for many concepts in science and engineering coursework (see Hibbeler, 2012; Serway & Jewett, 2008). However, several studies demonstrate that students are struggling to apply their knowledge of integration to subsequent courses (e.g. Beichner, 1994; Christensen & Thompson, 2010; Grundmeier, Hansen, & Sousa, 2006; Pollock, Thompson, & Mountcastle, 2007). This finding has led some researchers to begin to examine why students are having this difficulty. Sealey (2006) and Jones (2013) suggest that the “area under a curve” notion alone is not sufficient for understanding definite integrals. Thompson and Silverman (2008) promote the development of an “accumulation” conception of the integral in order to help students. Jones (under review) subsequently conducted a more thorough analysis of the anti-derivative, area under a curve, and summation interpretations of the definite integral by students in both mathematics and science contexts. The results demonstrate that the “summation” conception proved highly productive for understanding definite integrals that are either situated in a larger context or that contain variables representing physical quantities. By contrast, the study confirms that the “area under a 2014. In Oesterle, S., Liljedahl, P., Nicol, C., & Allan, D. (Eds.) Proceedings of the Joint Meeting 3 - 401 of PME 38 and PME-NA 36,Vol. 3, pp. 401-408. Vancouver, Canada: PME. Jones curve” and “values of an anti-derivative” conceptions are less productive in making sense of these types of contextualized definite integrals. While the findings do not imply that the area and anti-derivative ideas are not important (nor that they should not be learned) they do suggest that it is critical for students to have a robust and accessible summation conception of integration in their cognitive repertoire. Based on these results, it is important to ask the question: Are calculus students generally constructing their knowledge of the integral in a way that promotes the beneficial summation conception? This paper seeks to answer this question by investigating (a) how common each of the three conceptualizations are when a large sample of calculus students are asked to think about integration, and (b) whether standard ways of introducing Riemann sums are sufficient for a general student population to internalize the summation conception. THEORETICAL PERSPECTIVE Symbolic forms For this study, the manner in which students hold their knowledge of the integral is characterized through the lens of symbolic forms (Sherin, 2001). A symbolic form is a blend (Fauconnier & Turner, 2002) between a symbol template and a conceptual schema. The symbol template refers to the arrangement of the symbols in an equation [] or expression, such as []d [] , where each “box” can be filled in with symbols. The —[] conceptual schema is the meaning that underlies the symbols in the template. Jones (2013) documents students’ symbolic forms of the definite integral that are associated with the notions of area under a curve, values of an anti-derivative, and summations. Note that the students in the study regularly ascribed “anti-derivative” meanings to both indefinite and definite integrals. Furthermore, Jones describes a “deviant” of the typical Riemann sum conception that was dominant in some students’ thinking. These four symbolic forms aided the analysis of the student data by helping determine when students were drawing on each the area, anti-derivative, or summation conceptualizations of the integral. A brief description of each form is provided here. Area and perimeter: This symbolic form interprets each “box” in the symbol template as being one part of the perimeter of a shape in the (x-y) plane. The differential, “d[],” represents the “bottom” of the shape by dictating the variable that resides on the horizontal axis. This symbolic form is associated with the “area under a curve” notion. Function matching: This symbolic form interprets the integrand as having come from some “original function.” The original function became the integrand through a derivative, and the differential “d[]” indicates the variable with respect to which the derivative was taken. This form is associated with the “anti-derivative” conception. Adding up pieces: This form casts the differential as being a tiny piece of the domain, given by the limits of integration. Within each tiny piece, the quantities represented by the integrand and differential are multiplied to create a small amount of the resultant 3 - 402 PME 2014 Jones quantity. The integral symbol dictates an “infinite” summation that ranges over the domain. This form is related to the Riemann sum idea. Adding up the integrand: This is a “deviant” of the adding up pieces form. The key difference is that within each tiny piece, only the quantity represented by the integrand is added up. This resulting “total” from the integrand is then multiplied by the entire domain (length, area, or volume) to get the resulting quantity. This form fails to adequately describe the Riemann sum process, but is rooted in ideas of summations. Manifold view of knowledge Symbolic forms can be considered a subset of “cognitive resources” (Hammer, 2000), which are any piece of cognition that can be drawn on and employed as a unit, whether large or small. The main idea from cognitive resources that is used for this paper is the push away from a “unitary view” of concepts to a “manifold view” of knowledge (Hammer, Elby, Scherr, & Redish, 2005). The theory of resources argues that a “concept” such as the integral is comprised of many small and large elements—like rectangles, graphs, functions, ideas about summation, areas, limits, anti-derivative rules, and so forth—that are too complex to be considered a single entity. Thus, this study assumes that students can think about the integral by drawing on certain aspects of their “integral knowledge” while other aspects remain dormant. For example, a student may look at an integral and immediately think “area under a curve” without ever thinking about Riemann sums. This does not necessarily mean that the student does not have a Riemann sum conception in their cognition, but rather that the area conception is much more familiar and readily accessible to them. This has implications for learning integrals, since students need not only to “assimilate” a summation conception somewhere in their cognition, but that that conception needs to be created in a way that it is prevalent in their thinking to capitalize on its usefulness. METHODS Initial survey In order to investigate the prevalence of the area, anti-derivative, and summation conceptions of the definite integral, 150 students at two major colleges in the Western United States, who had successfully completed first-semester calculus, were recruited to participate in a survey that asked them open-ended questions about definite integrals. A χ2-test revealed no significant difference between the students at these two schools, in terms of the frequencies of the responses that were coded as belonging to the each conception of the integral used in this study (see below for more detail). This allows for the assumption that these students may be considered representative of the general calculus student population. The choice to use successful first-semester students is based on the fact that many key aspects of the integral are explored during the first semester: areas under curves, the Riemann integral definition, the Fundamental Theorem of Calculus (FTC), the Net Change Theorem, velocity/position applications, and anti-derivative techniques (including u-substitution). PME 2014 3 - 403 Jones To recruit students, several second-semester calculus courses were visited within the first two days of the semester in order to administer the survey. Students who had already taken second semester calculus (or the equivalent in high school) were asked not to complete the survey, to keep the focus of the study on students who had only successfully finished first-semester calculus. The students who participated were given fifteen minutes to complete the survey. Two of the four items from the survey form the focus for this paper. The first item b reads, “Explain in detail what fxdx() means. If you think of more than one way to —a describe it, please describe it in multiple ways. Please use words, or draw pictures, or write formulas, or anything else you want to explain what it means.” The item clearly asks the students to express any and all ways that they conceive of the integral and this instruction was reiterated to the students when the surveys were administered.
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